1,721,027 research outputs found
Eigenvalue estimates for the weighted laplacian on a riemannian manifold
Full text linkGiven a complete Riemannian manifold M and a smooth positive function w on M, let L = - Δ - ∇(log w) acting on L2(M, w dV). Generalizing techniques used in the case of the Laplacian, we obtain upper and lower bounds for the first non-zero eigenvalue of L, for M compact, and for the bottom of the spectrum, for M non-compact
Potential theory for manifolds with boundary and applications to controlled mean curvature graphs
No full textIn this paper we characterize the Neumann-parabolicity of manifolds with boundary in terms of a new form of the classical Ahlfors maximum principle and of a version of the so-called Kelvin–Nevanlinna–Royden criterion. The motivation underlying this study is to obtain new information on the geometry of graphs with prescribed mean curvature inside a Riemannian product of the type N × R. In this direction two kind of results will be presented: height estimates for constant mean curvature graphs parametrized over unbounded domains in a complete manifold, which extend results by A. Ros and H. Rosenberg valid for domains of R^2, and slice-type results for graphs whose superlevel sets have finite volume. Finally, the use of the Ahlfors maximum principle allows us to establish a connection between the Neumann-parabolicity and the Dirichlet-parabolicity commonly used in minimal surface theory. In particular, we will be able to give a deterministic proof of special cases of a result by R. Neel
and operator norm estimates for the complex time heat operator on homogeneous trees
No full textLet X be a homogeneous tree of degree greater than or equal to three. In this paper we study the complex time heat operator Hζ induced by the natural Laplace operator on X. We prove comparable upper and lower bounds for the Lp norms of its convolution kernel hζ and derive precise estimates for the Lp-Lr operator norms of Hζ for ζ belonging to the half plane Re ζ ≥ 0. In particular, when ζis purely imaginary, our results yield a description of the mapping properties of the Schrödinger semigroup on X. ©1998 American Mathematical Society
Eigenvalue estimates for the Laplacian with lower order terms on a compact Riemannian manifold.
No full textProc. Sympos. Pure Math., 54, Part 3, Amer. Math. Soc., Providence, RI, 1993
A lower bound for the spectrum of the Laplacian in terms of sectional and Ricci curvature.
No full textAsymptotic finite propagation speed for heat diffusion on certain Riemannian manifolds.
No full textAbstractUsing pointwise upper bounds recently obtained by the first author, we first show that the heat kernel ht(x, y) on a Riemannian symmetric space of the noncompact type GK is asymptotically concentrated in an annulus centered at y and moving to infinity with finite speed 2 ¦ϱ¦, ϱ being as usual the half sum of all positive roots of GK. In the higher rank case we prove moreover that heat not only concentrates in an annulus but also along the (K-orbit) of the ϱ-axis. By applying wave equation techniques developed by M. E. Taylor, we partially extend the above results to Riemannian manifolds with exponential volume growth and L2 spectrum of the Laplacian bounded away from 0. Some extensions to the vector bundle case are also considered
The Feller property on Riemannian manifolds
No full textThe asymptotic behavior of the heat kernel of a Riemannian manifold gives rise
to the classical concepts of parabolicity, stochastic completeness (or
conservative property) and Feller property (or -diffusion property).
Both parabolicity and stochastic completeness have been the subject of a
systematic study which led to discovering not only sharp geometric conditions
for their validity but also an incredible rich family of tools, techniques and
equivalent concepts ranging from maximum principles at infinity, function
theoretic tests (Khas'minskii criterion), comparison techniques etc... The
present paper aims to move a number of steps forward in the development of a
similar apparatus for the Feller property
On the form spectrum of the Laplacian on nonnegatively curved manifolds
No full textLet be a complete, noncompact Riemannian manifold with a pole, and let be a conformally related metric. We obtain conditions on the curvature of and on under which the Laplacian on -forms on has no eigenvalues.departmental bulletin pape
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